Surds are an expression in root form such as square root, cube root and other in a root symbol. A surd cannot be simplified to remove the root symbol. It does not have an exact decimal value and cannot be represented by a fraction. The decimal value just continues on and on to infinity, neither a terminating nor recurring decimal. It is also called an irrational number. 4 is not a surd as it can be simplified \(\sqrt{4}\) = 2. \(\sqrt{2}\) is an irrational numbers. \(\sqrt{2} \) =1.41421356… and does not have an exact decimal form.

Manipulating surds using operations, there are rules to follow in order to perform a mathematical operation. Let’s first learn a way to simplify a surd.

Example:Simplify \(\sqrt{200}\) Think of a factor of 200, it can be 100 multiplied by 2.
\(\sqrt{100 \cdot 2}\) The square root of one hundred is 10
\(10\sqrt{2}\) is the simplified form of \(\sqrt{200}\).

Multiplying SurdsMultiplying surds is simply combining surds. However, if like surds are multiplied, an answer is a rational number.
Example 1: \(\sqrt{2} \sqrt{ (2)}\)
\(\sqrt{2} \sqrt{ (2)}\) They are like surds since their radicand is both \(\sqrt{2}\).
\(\sqrt{2 \cdot 2}\)
\(\sqrt{4}\)The answer is always in simplest form:

Adding and Subtracting SurdsA surd that has the same radicand can be added or subtracted. Before adding or subtracting surds, simplify the expressions.
Example: \(\sqrt{5}+ 3 \sqrt{5}\)
\(\sqrt{5}+ 3 \sqrt{5}\) Notice that they have same radicand which is \(\sqrt{5}\).
\(\sqrt{5}+ 3 \sqrt{5}\) Add like surds.
\(4 \sqrt{5}\)

Rationalising the DenominatorA surd is cannot be a denominator. If the denominator is a surd, it is not the simplest form.
To rationalise a denominator, we need to clear the radicals.
1. Multiply the numerator and the denominator that make the denominator a perfect root.
2. Simplify the radicals.